Functional notation and transformations is always tricky to teach and understand. GCSE students will meet this in Year 11. In general, transformations applied after the function are more easily understood:

or

Something very un-intuitive happens when the transformation is applied to the argument of the function:

or

with things stretching when they look like they should be compressing and other things moving the wrong way.

There’s always space on the inter-web for another proof of Pythagoras’s Theorem. Here’s one that uses the following equal areas property of parallelograms.

This kind of area chopping and shape translation is a feature of Euclidean geometry and our senses support it’s veracity at the order of size of the classroom.

The squares on the sides of a right-angle triangle set up a system of parallel lines which can then be used to demonstrate the Theorem using the above equal areas property.

The thread does not stop here though. Taking the parallel line structure which makes this work we get a generalisation of Pythagoras to non-right-angled triangles with the area of the parallelogram on the longest side being the sum of the areas of those constructed on the other two sides.

Bishop Berkeley writes this attack on the apparent supernatural reasoning involved in calculus. The infidel was probably Halley (of comet fame) or Newton.

If pupils find the subject difficult to understand at school, and teachers find it difficult to teach, then the reason may be articulated in this book by the great man.

Quotes include:

“Now to conceive a Quantity infinitely small, that is, infinitely less than any sensible or imaginable Quantity, or any the least finite Magnitude, is, I confess, above my Capacity. But to conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply’d infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever”

and.

“They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?”

Some sympathy for the thesis is gained by Berkeley’s examination of tangent reasoning:

“Therefore the two errors being equal and contrary destroy each other; the first error of defect being corrected by a second error of excess. ……. If you had committed only one error, you would not have come at a true Solution of the Problem. But by virtue of a twofold mistake you arrive, though not at Science, yet at Truth. For Science it cannot be called, when you proceed blindfold, and arrive at the Truth not knowing how or by what means.”

The student of sixth form level mathematics who is eager to see how this is resolved must continue their path to mathematical enlightenment by studying the Analysis of Cauchy, Riemann and Weierstrass.

I use this mostly as a resource for my own teaching; using it I can lay my hands of exam questions and solutions of different types very quickly. My students and pupils find it useful too because it shows how to do exam questions using the techniques taught in class; there is an important difference with mark schemes here.

The UK mathematics exam system is in a state of flux at the moment. I see this is as a great opportunity to review and refresh all my teaching material. I look forward to generating new material to support a revised school exam system.

Logarithmic spirals frequently occur in nature. Is this such a manifestation?

The general equation for a logarithmic spiral is as follows.

Changing the variables and produces spirals of different qualities. The is really an enlargement scale factor but the controls how the spiral grows per revolution.

If is more representative of an ammonite then seems to be our Greenland depression.

Wouldn’t it be nice to take the equations for atmospheric fluid dynamics and show this explicitly. Unfortunately this is beyond the scope of this blog: here we have circular motion on the surface of a rotating sphere in an elliptical orbit … . The way to go with this would be to take the basic baratropic equations and then perform a scale analysis to disregard ‘small’ terms. At this point the pure mathematician get frustrated with approximations and goes into a sulk. We are left with contemplating the beautiful images though…

This method for finding the centre, and by extension the equation, of a circle given three non-colinear points, brings the ancient textbook master Euclid onto Descartes’ coordinate plane and right into the 21 century classroom.

The sine function starts life in a triangle (year 9?) then becomes intimate with a circle at A level, as y-coord of a point rotating on a unit circle. Later it is about infinite series or, even, differential equations.

Have got absorbed by the National Cipher Challenge. Not sure what this animation varying the multiplier in an ‘Affine Shift’ add 5. Functional grid shows transition from Caesar Shift 5, multiply by 1, through to multiply by 13, which is a useless cipher. In fact any multiplier not co-prime with 26 can be seen to be not one-one, and therefore useless too. Modular arithmetic, straight line graphs, ciphers, dotty patterns, ….